How do you use the Product Rule to find the derivative of #y = x(x^2 - 2x + 1)^4#?

Question

Cameron Alexander · Accepted Answer

#y^' = (9x-1) * (x-1)^7#

The product rule allows you to differentiate functions that can be written as the product of two other functions

#y = f(x) * g(x)#

by using this formula

#color(blue)(d/dx(y) = f^'(x) * g(x) + f(x) * g^'(x)#

In your case, you can write #y# as

#y = x * (x^2 - 2x + 1)^4 = x * [(x-1)^2]^4 = x * (x-1)^8#

This means that its derivative can be found by using the product rule

#y^' = [d/dx(x)] * (x-1)^8 + x * d/dx(x-1)^8#

#y^' = 1 * (x-1)^8 + x * 8 * (x-1)^7#

#y^' = (x-1)^7(x-1 + 8x) = color(green)( (9x-1) * (x-1)^7)#

Samantha Adkison · Answer

undefined To find the derivative of $ y = x(x^2 - 2x + 1)^4 $ using the Product Rule, follow these steps:

1. Identify the functions $ f(x) $ and $ g(x) $ in the form $ f(x) \cdot g(x) $.
   $ f(x) = x $ and $ g(x) = (x^2 - 2x + 1)^4 $.

2. Apply the Product Rule formula: $ (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x) $.

3. Find the derivatives of $ f(x) $ and $ g(x) $:
   $ f'(x) = 1 $ and $ g'(x) = 4(x^2 - 2x + 1)^3 \cdot (2x - 2) $.

4. Substitute these derivatives into the Product Rule formula:
   $ y' = 1 \cdot (x^2 - 2x + 1)^4 + x \cdot 4(x^2 - 2x + 1)^3 \cdot (2x - 2) $.

5. Simplify the expression:
   $ y' = (x^2 - 2x + 1)^4 + 4x(x^2 - 2x + 1)^3 \cdot (2x - 2) $.